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The main points which I found very interesting are: Time Series Analysis, Sequence of Values, Random Variable, Discrete Time Series, Hydrologic Time Series, Stochastic Components, Continuous Time Series, Time Scale, Probabilistic Behavior, Time Average for Realization
Typology: Study notes
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time
stochastic components
t
= d t
3
t
x t
Long term mean
4
t
x t
Stochastic + Trend
t
x
t
Stochastic + Periodic
t
x t
Stochastic + Jump
t
x
t
Stochastic
t
= d t
specified time ‘t’
do not change with time.
t
is same as that of X t
v t
6
Time average for a realization
n is no. of observations
Ensemble average at time t
m is no. of realizations
7
1
1
1
n
j
j
X t
n
=
1
m
i
i
t
=
∑
t
t
1
Realization-
t
Realization-
t
2
t
t
m
Realization-m
t 1 t 1 t 1
t
and X t+τ
9
2
0
0
cov ,
cov ,
t t k
t t k
k
X X
t t k k
X
ρ
σ σ
γ
σ γ
ρ
2
0
cov , k t t k
t t k
X
γ
μ μ
γ σ
If process is stationary
t t k
X X
σ σ
stochastic process
10
k
ρ k
Correlogram
n
by γ o
, we get the auto
correlation matrix Ρ n
n
is symmetric and +ve definite matrix
12
1 2 1
1 1 2
2
0
1 2
n
n
n
n
n n
ρ ρ ρ
ρ ρ ρ
ρ
γ
ρ ρ
−
−
− −
n x n
n
is +ve definite
13
1
1
2
1
1
ρ
ρ
ρ
ρ
k
If it is purely stochastic (random) series,
ρ
k
= 0, v k = 1, 2, 3,……..
r
k
= may not be zero (because r k
is a sample estimate)
15
k
k
r k
Correlogram
For a random series
16
k
-z +z
k
r
k
Statistically insignificant
z = 1.
For a random series
18
mean = 1075/
Variance,
x
2
1
0
1 10 1
n
t
t
x x
c
n
=
−
= = =
− −
1
1
1
1
10
n
t t
t
x x x x
c
n
−
=
− −
= = =
1
1
0
c
r
c
= = =
Obtain correlogram for 40 uniformly distributed random
numbers
19
S.No. Data S.No. Data S.No. Data S.No. Data
(^1 98 11 73 21 25 31 )
(^2 69 12 36 22 49 32 )
(^3 30 13 11 23 73 33 )
(^4 50 14 54 24 38 34 )
(^5 93 15 31 25 14 35 )
(^6 1 16 74 26 4 36 )
(^7 66 17 23 27 87 37 )
(^8 99 18 88 28 99 38 )
(^9 76 19 82 29 69 39 )
(^10 65 20 92 30 57 40 )
21
k
r
k
40
40
−
Purely stochastic process
22
N
N
−
Statistically insignificant
k
r
k
k
r k
Periodic process
